Bijective proof and generalization of Siladi cs partition theorem - PowerPoint PPT Presentation

A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Bijective proof and generalization of Siladi´ c’s partition theorem Isaac KONAN IRIF, Paris Diderot Alea Days 2019, March 21st 1/20 Isaac KONAN Siladi` c’s partition theorem

A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Overview 1 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement 2 Generalization of Siladi´ c’s theorem Infinite set of primary colors Generalized theorem 3 Bijective map 4 Further analysis on higher degrees 2/20 Isaac KONAN Siladi` c’s partition theorem

A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Partitions and Rogers-Ramanujan type Identities Generalization of Siladi´ c’s theorem Siladi´ c’s partition theorem Bijective map Dousse’s refinement Further analysis on higher degrees Colored integer partitions Finite “decreasing” sequence of colored positive integers λ = ( λ 1 , . . . , λ s ). Example : λ = (4 , 2 , 1 , 1) 3/20 Isaac KONAN Siladi` c’s partition theorem

A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Partitions and Rogers-Ramanujan type Identities Generalization of Siladi´ c’s theorem Siladi´ c’s partition theorem Bijective map Dousse’s refinement Further analysis on higher degrees Colored integer partitions Finite “decreasing” sequence of colored positive integers λ = ( λ 1 , . . . , λ s ). Example : λ = (4 , 2 , 1 , 1) 3/20 Isaac KONAN Siladi` c’s partition theorem

A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Partitions and Rogers-Ramanujan type Identities Generalization of Siladi´ c’s theorem Siladi´ c’s partition theorem Bijective map Dousse’s refinement Further analysis on higher degrees Colored integer partitions Finite “decreasing” sequence of colored positive integers λ = ( λ 1 , . . . , λ s ). Example : λ = (4 , 2 , 1 , 1) • Parts’length : 4 , 2 , 1 , 1 • Color sequence : c ( λ ) = � i c i = red · blue · blue · green 3/20 Isaac KONAN Siladi` c’s partition theorem

A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Partitions and Rogers-Ramanujan type Identities Generalization of Siladi´ c’s theorem Siladi´ c’s partition theorem Bijective map Dousse’s refinement Further analysis on higher degrees Colored integer partitions Finite “decreasing” sequence of colored positive integers λ = ( λ 1 , . . . , λ s ). Example : λ = (4 , 2 , 1 , 1) • Parts’length : 4 , 2 , 1 , 1 • Color sequence : c ( λ ) = � i c i = red · blue · blue · green • Size : | λ | = � i λ i = 4 + 2 + 1 + 1 = 8 3/20 Isaac KONAN Siladi` c’s partition theorem

A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Partitions and Rogers-Ramanujan type Identities Generalization of Siladi´ c’s theorem Siladi´ c’s partition theorem Bijective map Dousse’s refinement Further analysis on higher degrees Colored integer partitions Finite “decreasing” sequence of colored positive integers λ = ( λ 1 , . . . , λ s ). Example : λ = (4 , 2 , 1 , 1) • Parts’length : 4 , 2 , 1 , 1 • Color sequence : c ( λ ) = � i c i = red · blue · blue · green • Size : | λ | = � i λ i = 4 + 2 + 1 + 1 = 8 • Consecutive differences (increments) λ i − λ i +1 : 2 , 1 , 0. 3/20 Isaac KONAN Siladi` c’s partition theorem

A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Partitions and Rogers-Ramanujan type Identities Generalization of Siladi´ c’s theorem Siladi´ c’s partition theorem Bijective map Dousse’s refinement Further analysis on higher degrees Colored integer partitions Finite “decreasing” sequence of colored positive integers λ = ( λ 1 , . . . , λ s ). Example : λ = (4 , 2 , 1 , 1) • Parts’length : 4 , 2 , 1 , 1 • Color sequence : c ( λ ) = � i c i = red · blue · blue · green • Size : | λ | = � i λ i = 4 + 2 + 1 + 1 = 8 • Consecutive differences (increments) λ i − λ i +1 : 2 , 1 , 0. For any set of partitions A , enumeration according to partitions’ � c ( λ ) q | λ | . size and color sequence : GF A ( q ) = λ ∈ A 3/20 Isaac KONAN Siladi` c’s partition theorem

A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Partitions and Rogers-Ramanujan type Identities Generalization of Siladi´ c’s theorem Siladi´ c’s partition theorem Bijective map Dousse’s refinement Further analysis on higher degrees Rogers-Ramanujan type identities Theorem (RR1919) Same cardinalities for sets of partitions of size n with: • consecutive differences at least 2 , • parts congruents to ± 1 mod 5 . 4/20 Isaac KONAN Siladi` c’s partition theorem

A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Partitions and Rogers-Ramanujan type Identities Generalization of Siladi´ c’s theorem Siladi´ c’s partition theorem Bijective map Dousse’s refinement Further analysis on higher degrees Rogers-Ramanujan type identities Theorem (RR1919) Same cardinalities for sets of partitions of size n with: • consecutive differences at least 2 , • parts congruents to ± 1 mod 5 . Rogers-Ramanujan type identity : equality between two sets of partitions with conditions on respectively : • consecutive differences, • parts’congruences . 4/20 Isaac KONAN Siladi` c’s partition theorem

A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Partitions and Rogers-Ramanujan type Identities Generalization of Siladi´ c’s theorem Siladi´ c’s partition theorem Bijective map Dousse’s refinement Further analysis on higher degrees Rogers-Ramanujan type identities Theorem (RR1919) Same cardinalities for sets of partitions of size n with: • consecutive differences at least 2 , • parts congruents to ± 1 mod 5 . Rogers-Ramanujan type identity : equality between two sets of partitions with conditions on respectively : • consecutive differences, • parts’congruences . Example of Euler distinct-odd (1748) : distincts parts ≡ odd parts. 4/20 Isaac KONAN Siladi` c’s partition theorem

A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Partitions and Rogers-Ramanujan type Identities Generalization of Siladi´ c’s theorem Siladi´ c’s partition theorem Bijective map Dousse’s refinement Further analysis on higher degrees Siladi´ c’s partition theorem (2002) Theorem The number of partitions ( λ 1 , . . . , λ s ) of an integer n into parts λ i different from 2 , such that λ i − λ i +1 ≥ 5 and with additional conditions for 5 ≤ λ i − λ i +1 ≤ 8 according to the table below : λ i − λ i +1 λ i + λ i +1 mod 16 λ i − λ i +1 λ i + λ i +1 mod 16 , 5 ± 3 6 0 , ± 4 , 8 7 ± 1 , ± 5 , ± 7 8 0 , ± 2 , ± 6 , 8 is equal to the number of partitions of n into distinct odd parts. 5/20 Isaac KONAN Siladi` c’s partition theorem

A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Partitions and Rogers-Ramanujan type Identities Generalization of Siladi´ c’s theorem Siladi´ c’s partition theorem Bijective map Dousse’s refinement Further analysis on higher degrees Siladi´ c’s partition theorem (2002) Theorem The number of partitions ( λ 1 , . . . , λ s ) of an integer n into parts λ i different from 2 , such that λ i − λ i +1 ≥ 5 and with additional conditions for 5 ≤ λ i − λ i +1 ≤ 8 according to the table below : λ i − λ i +1 λ i + λ i +1 mod 16 λ i − λ i +1 λ i + λ i +1 mod 16 , 5 ± 3 6 0 , ± 4 , 8 7 ± 1 , ± 5 , ± 7 8 0 , ± 2 , ± 6 , 8 is equal to the number of partitions of n into distinct odd parts. 5/20 Isaac KONAN Siladi` c’s partition theorem

A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Partitions and Rogers-Ramanujan type Identities Generalization of Siladi´ c’s theorem Siladi´ c’s partition theorem Bijective map Dousse’s refinement Further analysis on higher degrees Siladi´ c’s partition theorem (2002) Theorem The number of partitions ( λ 1 , . . . , λ s ) of an integer n into parts λ i different from 2 , such that λ i − λ i +1 ≥ 5 and with additional conditions for 5 ≤ λ i − λ i +1 ≤ 8 according to the table below : λ i − λ i +1 λ i + λ i +1 mod 16 λ i − λ i +1 λ i + λ i +1 mod 16 , 5 ± 3 6 0 , ± 4 , 8 7 ± 1 , ± 5 , ± 7 8 0 , ± 2 , ± 6 , 8 is equal to the number of partitions of n into distinct odd parts. Example of n = 15 5/20 Isaac KONAN Siladi` c’s partition theorem